# Lecture 20: Fourier Transform, The Wave Equation, and Weak Solutions

In this lecture we defined the Fourier transform of a tempered distribution and proved that it satisfies the same good properties as the Fourier transform on Schwartz functions. We then used it to solve the IVP for the wave equation:

$$ (\partial_t -\Delta_x)u = 0\quad u(0,x)=\phi(x)\quad \partial_t u(0,x)=\psi(x).$$

The development followed Andras Vasy's notes http://math.stanford.edu/~andras/172-5.pdf (see page 8 onwards) very closely. The point is that even though the solution $u(x,t)$ is eventually a $C^{\infty}$ function, it is easier to pass through the space of tempered distributions to find this solution.